18 research outputs found

    The codegree threshold of K4−K_4^-

    Get PDF
    The codegree threshold ex2(n,F)\mathrm{ex}_2(n, F) of a 33-graph FF is the minimum d=d(n)d=d(n) such that every 33-graph on nn vertices in which every pair of vertices is contained in at least d+1d+1 edges contains a copy of FF as a subgraph. We study ex2(n,F)\mathrm{ex}_2(n, F) when F=K4−F=K_4^-, the 33-graph on 44 vertices with 33 edges. Using flag algebra techniques, we prove that if nn is sufficiently large then ex2(n,K4−)≤(n+1)/4\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration GG, there is a quasirandom tournament TT on the same vertex set such that GG is close in the edit distance to the 33-graph C(T)C(T) whose edges are the cyclically oriented triangles from TT. For infinitely many values of nn, we are further able to determine ex2(n,K4−)\mathrm{ex}_2(n, K_4^-) exactly and to show that tournament-based constructions C(T)C(T) are extremal for those values of nn.Comment: 31 pages, 7 figures. Ancillary files to the submission contain the information needed to verify the flag algebra computation in Lemma 2.8. Expands on the 2017 conference paper of the same name by the same authors (Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413

    Limits of Order Types

    Get PDF
    The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs. We first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly. We next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester\u27s problem on the probability that k random points are in convex position

    Rainbow triangles in three-colored graphs

    Get PDF
    Erdős and Sós proposed the problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a) + F(b) + F(c) + F(d) + abc + abd + acd + bcd, where a + b + c + d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k ≥ 0. These results imply that lim F(n)/((n)(3)) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments
    corecore